Michael Cohen taped a face-to-face conversation with Donald Trump in which the two men discussed buying the rights to a former Playboy Playmate’s story that she had an affair with Mr. Trump more than a decade ago, according to people familiar with the matter.

The conversation between the then-GOP presidential nominee and his longtime personal lawyer took place in September 2016, the people said. That was two months before the election and a month after American Media Inc., the publisher of the National Enquirer, bought the...